Basic Definitions

The starting point is a smooth system of differential equations
with an equilibrium (rest point) at the origin, expanded as a power series
\[
\dot x = Ax + a_1(x) + a_2(x) +\cdots,
\]
where \(x\in{\mathbb R}^n\) or \({\mathbb C}^n\ ,\) \(A\) is an
\(n\times n\) real or complex matrix, and \(a_j(x)\) is a homogeneous polynomial
of degree \(j+1\) (for instance, \(a_1(x)\) is quadratic).
The expansion is taken to some finite order \(k\) and truncated there,
or else is taken to infinity but is treated formally (the convergence or divergence of the series is ignored).
The purpose is to obtain an approximation to the (unknown) solution of the original system, that will be valid over an extended range in time.
The linear term \(Ax\) is assumed to be already in the desired normal form,
usually the Jordan or a real canonical form. A transformation to new variables \(y\) is
applied, having the form
\[
x=y+u_1(y)+u_2(y)+\cdots,
\]
where \(u_j\) is homogeneous of degree \(j+1\ .\)
This results in a new system
\[
\dot y = Ay + b_1(y) + b_2(y) +\cdots,
\]
having the same general form as the original system. The goal is to make a
careful choice of the \(u_j\ ,\) so that the \(b_j\) are "simpler" in
some sense than the \(a_j\ .\) "Simpler" may mean only that some
terms have been eliminated, but in the best cases one hopes to
achieve a system that has additional symmetries that were not
present in the original system. (If the normal form possesses a
symmetry to all orders, then the original system had a hidden
approximate symmetry with transcendentally small error.)

Among many historical references in the
development of normal form theory, two significant ones are Birkhoff (1996) and Bruno (1989).
As the Birkhoff reference shows, the early stages of the theory
were confined to Hamiltonian systems, and the normalizing
transformations were canonical (now called symplectic). The Bruno
reference treats in detail the convergence and divergence of
normalizing transformations.

An Example

A basic example is the nonlinear oscillator with \(n=2\) and
\[
A=\left[
\begin{matrix}
0 & -1 \\ 1 & 0
\end{matrix}
\right].
\]
In this case it is possible (no matter what the original \(a_j\) may
be) to achieve \(b_j=0\) for \(j\) odd and to eliminate all but two
coefficients from each \(b_j\) with \(j\) even. More precisely,
writing \(r^2=y_1^2+y_2^2\ ,\) a normal form in this case is
\[
\dot y = Ay + \sum_{i=1}^{\infty} \alpha_ir^{2i}y + \beta_ir^{2i}Ay
\ .\]
In polar coordinates this becomes
\[
\dot r = \alpha_1 r^3 + \alpha_2r^5+\cdots
\]
\[
\dot\theta = 1 + \beta_1r^2 + \beta_2r^4+\cdots
\ .\]
The first nonzero \(\alpha_i\) determines the stability of the origin,
and the \(\beta_i\) control the dependence of frequency on amplitude.
Also the normalized system has achieved symmetry (more technically,
equivariance) under rotation about the origin. Although the
classical (or level-one) approach to normal forms stops with
the form obtained above for this example, it is important to note that neither
the coefficients \(\alpha_i\) and \(\beta_i\) in the equation, nor the
transformation terms \(u_j\) used to achieve the equation, are uniquely
determined by the original \(a_j\ .\) In fact, by a more careful
choice of the \(u_j\ ,\) it is possible to put the nonlinear oscillator
into a hypernormal form (also called a unique, higher-level, or simplest normal form) in which all but finitely many of the coefficients \(\alpha_i\) and \(
\beta_i\) are zero.
Hypernormal forms are difficult to calculate, and from here on we speak only of classical normal forms.

Asymptotic Consequences of Normal Forms

For some systems, the normal form (truncated at a given degree) is
simple enough to become solvable. In this case it is of interest
to ask whether this solution gives rise to a good approximation (an
asymptotic approximation in some specific sense) to a solution of
the original equation (say, with the same initial condition). The
answer is "sometimes yes". ("Gives rise to" means that the
solution of the truncated normal form usually must be fed back through
the transformation to normal form.) Some popular
books, such as Nayfeh (1993), present the subject entirely
from this point of view, without proving any error estimates or
noticing that there are cases in which asymptotic validity cannot
hold. Several theorems and open questions in this regard are given
in chapter 5 of Murdock (2003). The most basic theorem states that an asymptotic error estimate with respect to a small parameter holds if (a) the parameter is introduced correctly, (b) the matrix of the linear term is semisimple (see below) and has all its eigenvalues on the imaginary axis, and (c) the semisimple normal form style (see below) is used. Although the asymptotic use of
normal forms is important when it is true, and has many practical
applications, the primary importance of normal forms is as a
preparatory step towards the study of qualitative dynamics,
unfoldings, and bifurcations.

Geometrical Consequences of the Normal Form

It has already been pointed out that a normal form can decide
stability questions and establish hidden symmetries. Computing the
normal form up to degree \(k\) also automatically computes (to degree
\(k\)) the stable, unstable, and center manifolds, the
center manifold reduction, and the fibration of the center-stable and
center-unstable manifolds over the center manifold. The common
practice of computing the center manifold reduction first, and then
computing the normal form only for this reduced system, seems to
save work but loses many of these results. See chapter 5 of Murdock (2003).

On occasion, the truncation of a normal form produces a simple
system that is topologically equivalent to the original system in
a neighborhood of the equilibrium, called topological normal form.
For instance, in the example above, truncating after the first nonvanishing \(\alpha_i\) will accomplish
this, but if all \(\alpha_i\) are zero, the topological behavior is
probably determined by a transcendentally small effect that is not
captured by the normal form.

Normal forms are important for determining bifurcations of a
system, but this requires the inclusion of unfolding parameters.

The Homological Equation and Normal Form Styles

In the general case, we define the Lie derivative operator
\(L_A\) associated with the matrix \(A\) by \((L_A v)(x)=v'(x)Ax-Av(x)\ ,\)
where \(v\) is a vector field and \(v'\) is its matrix of partial derivatives.
Then \(L_A\) maps the vector space \(\mathcal{V}_j\) of
homogeneous vector fields of degree \(j+1\) into itself. The
relation between the \(a_j\ ,\) \(b_j\ ,\) and \(u_j\) is determined
recursively by the homological equations
\[
L_A u_j = K_j - b_j
\ ,\]
where \(K_1=a_1\) and \(K_j\) equals \(a_j\) plus a correction term
computed from \(a_1,\dots,a_{j-1}\) and \(u_1,\dots,u_{j-1}\ .\)
Let \(\mathcal{N}_j\) be any choice of a complementary subspace to the image of
\(L_A\) in \(\mathcal{V}_j\ ;\) then it is possible to choose the \(u_j\) so that
each \(b_j\in \mathcal{N}_j\ .\) (Take \(b_j=P_j K_j\ ,\) where
\(P_j:\mathcal{V}_j\rightarrow\mathcal{N}_j\) is the projection map, and note that the
homological equation can be solved, nonuniquely, for \(u_j\ .\)) The
choice of \(\mathcal{N}_j\) is called a normal form style, and
represents the preference of the user as to what is considered "simple". The purpose of this procedure is to ensure that the higher-order correction terms, \(u_j\ ,\) are bounded, so that the approximation to the solution, \(x(t)\ ,\) is valid over an extended range in time.

The theory breaks into two cases according to whether \(A\) is
semisimple (diagonalizable) or not. The semisimple case,
illustrated by the nonlinear oscillator above, is the easiest, and
there is only one useful style (in which \(\mathcal{N}_j\) is the kernel of
\(L_A\)), ultimately due to Poincaré. It is easy to describe the
semisimple normal form if \(A\) is diagonal with diagonal entries
\(\lambda_1,\dots,\lambda_n\) (which usually requires introducing complex
variables with reality conditions): The \(r\)th equation (for
\(\dot y_r\)) of the normalized system will contain only
monomials \(y_1^{m_1}\cdots y_n^{m_n}\) satisfying
\[
m_1\lambda_1+\cdots+m_n\lambda_n-\lambda_r=0
\ .\]
Such monomials are called resonant because for pure
imaginary eigenvalues, this equation becomes a resonance among
frequencies in the usual sense. An elementary treatment of normal
forms in the semisimple case only is by Kahn and Zarmi (1998).

The Nonsemisimple Case

In the nonsemisimple case there are two important styles, the
inner product normal form, originally due to Belitskii but
popularized by Elphick et al. (1987), and the sl(2) normal form due to
Cushman and Sanders. In the inner product style, \(\mathcal{N}_j\) is the
kernel of \(L_{A^*}\ ,\) \(A^*\) being the adjoint or conjugate transpose
of \(A\ .\) In the sl(2) style, \(\mathcal{N}_j\) is the kernel of an operator
defined from \(A\) using the theory of the Lie algebra sl(2). The
inner product style is more popular at this time, but the sl(2)
style has a much richer mathematical structure with deep
connections to sl(2) representation theory and to the
classical invariant theory of Cayley, Sylvester and others.
Because of this the sl(2) style has computational algorithms that
are not available for the inner product style. There is also a
simplified normal form style that is derived from the inner
product style by changing the projection.

A modern introduction to normal form theory, containing all the
styles mentioned here with references and historical remarks, may
be found in the monograph by Murdock (2003). Some more recent developments are contained in the last few chapters of Sanders, Verhulst, and Murdock (2007).